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For graphs $G$ and $H$, let $G overset{mathrm{rb}}{{longrightarrow}} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, v{S}kori{c} and Steger [J. Combin. Theory Ser. B 124 (2017),1-38], we determine the threshold for $G(n,p) overset{mathrm{rb}}{{longrightarrow}} C_ell$ for cycles $C_ell$ of any given length $ell geq 4$.
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The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.
For graphs $G$ and $H$, let $G {displaystylesmash{begin{subarray}{c} hbox{$tinyrm rb$} longrightarrow hbox{$tinyrm p$} end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that, for every graph $H$, an asymptotic upper bound for the threshold function $p^{rm rb}_H=p^{rm rb}_H(n)$ of this property for the random graph $G(n,p)$ is $n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum $2$-density of $H$. Extending a result of Nenadov, Person, v{S}koric, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p^{rm rb}_{K_k}$ for $kgeq 5$. Furthermore, we show that $p^{rm rb}_{K_4} = n^{-7/15}$.
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper.
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turan number $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ $bigg($ respectively, $text{ex}(K_{n_{1},n_{2},n_{3}},{C_{3}, C_{4}^{text{multi}}})$ $bigg)$ is the maximum number of edges in a graph $Gsubseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{text{multi}}$ $bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{text{multi}}$ $bigg)$. We call a $C^{multi}_4$ rainbow if all four edges of it have different colors. The ant-Ramsey number $text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ is the maximum number of colors in an edge-colored of $K_{n_{1},n_{2},n_{3}}$ with no rainbow $C_{4}^{text{multi}}$. In this paper, we determine that $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})=n_{1}n_{2}+2n_{3}$ and $text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})=text{ex}(K_{n_{1},n_{2},n_{3}}, {C_{3}, C_{4}^{text{multi}}})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}ge n_{2}ge n_{3}ge 1.$
Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $ colors, $G$ contains a monochromatic copy of $ H $. Haxell, Kohayakawa and L uczak showed that the size-Ramsey number of a cycle $ C_n $ is linear in $ n $ i.e. $ hat{R}(C_n, r) leq c_rn $, for some constant $ c_r $. Their proof, however, is based on the Szemeredis regularity lemma and so no specific constant $ c_r $ is known. Javadi, Khoeini, Omidi and Pokrovskiy gave an alternative proof for this result which avoids using of the regularity lemma. Indeed, they proved that if $ n $ is even, then $ c_r $ is exponential in $ r $ and if $ n $ is odd, then $ c_r $ is doubly exponential in $ r $. oindent In this paper, we improve the bound $c_r$ and prove that $c_r$ is polynomial in $r$ when $n$ is even and is exponential in $r$ when $n$ is odd. We also prove that in the latter case, it cannot be improved to a polynomial bound in $r$. More precisely, we prove that there are some positive constants $c_1,c_2$ such that for every even integer $n$, we have $c_1r^2nleq hat{R}(C_n,r)leq c_2r^{120}(log^2 r)n$ and for every odd integer $n$, we have $c_1 2^{r}n leq hat{R}(C_n, r)leq c_22^{16 r^2+2log r}n $.
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